Higher Order Partial Derivatives

    The partial derivative is a function, so it is possible to take the partial derivative of a partial derivative. This is very much like taking the second derivative of a function of one variable if we take two consecutive partial derivatives with respect to the same variable, and the resulting derivative is called the second-order partial derivative with respect to that variable. However, we can also take the partial derivative with respect to one variable and then take a second partial derivative with respect to a different variable, producing what is called a mixed second-order partial derivative.

Definition (Higher Order Partial Derivatives) The higher-order partial derivatives for a function of two variables are denoted as


and


and the mixed second partial derivatives are denoted as


and

Example (Higher Order Partial Derivatives) Compute the following partial derivatives.

(a) Find the second partial derivatives of .

    Solution. The first partial derivatives are

     and     

Therefore,

Proposition (Clairaut's Theorem) If the function has mixed second-order partial derivatives and , that are continuous on an open disk containing , then

    Proof. For small values of with , consider the difference



Notice that if we let


then


By the Mean Value Theorem, there is a number between and such that



Applying the Mean Value Theorem again, this time to we get a number between and such that



Combining these equations, we obtain



If , then , so the continuity of at gives



Similarly, by writing



and using the Mean Value Theorem twice and the continuity of at , we obtain  



It follows that as desired.

Example (Computing Partial Derivatives)  Compute the following partial derivatives.  

(a) Verify that the function satisfies the wave equation

    Solution. We find that









and therefore,

(b) Verify that the function is a solution of Laplace's equation

    Solution. We find that  









and therefore,


(c) Verify that the functions and satisfy the Cauchy-Riemann equations
    

    
    Solution. We find that
    

and


(d) Verify that the function is a solution of the three-dimensional Laplace equation  

    Solution. We compute



    

    

  Therefore,  




(e) Show that the function is a solution of the equation

  

    Solution.  We compute

    

    

    

  and so
   
   
  as desired.

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